<p>Which are correct for relation <span class="math-inline">\(R\)</span> on <span class="math-inline">\(\mathbb{R}\)</span>?</p>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>(A) Not symmetric (|x|>|y| does not imply |y|>|x|), not transitive (small differences accumulate). ✓</p><p>(B) Not transitive: |x-y|=0.8, |y-z|=0.8 can give |x-z|=1.6>1. ✗</p><p>(C) Reflexive (|x|-|x|=0≤1). Not symmetric: (2,0)∈R but (0,2)∉R. ✓</p><p>(D) Reflexive and symmetric. ✓</p><p><strong>Answer: (A),(C),(D)</strong></p><div class="trap-box"><strong>Trap:</strong> One counterexample suffices to disprove transitivity.</div><div class="key-concept"><strong>Key Concept:</strong> Test each property by definition; counterexample beats general argument for negation</div></div>
Correct Answer: A,C,D